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Gregorian leap day: Difference between revisions

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Gregorian leap day is a rank-2 temperament which is produced by temperament-merging 97edo, which has the cardinality of leap years in Gregorian calendar's cycle, and 400edo, the whole duration of the cycle.
{{Novelty}}
== Theory ==
{{Distinguish|Leapday}}
400 is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. Because the generator is mapped to 18/17, the temperament can be seen as an octavated version of [[18/17 equal-step tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].
'''Gregorian leap day''' is a [[rank-2 temperament]] which is produced by temperament-merging 97edo, which has the cardinality of leap years in Gregorian calendar's cycle, and 400edo, the whole duration of the cycle.
 
400 is the number of years in the Gregorian calendar's leap cycle. They are not spread evenly, but if they were, this would produce a scale with a 33\400 generator which is associated to [[18/17]] and [[55/52]], three of which make [[19/16]]. The optimal tuning is very close to 18/17, which makes it very similar to [[Galilei's tuning]]. Gregorian leap day has mos of size 12, 13, 25, 37, 49, 61, 73, and 97. [[1L 11s]] mos of this temperament is a barely noticeable circulating temperament for [[12edo]].


In the 7-limit, temperament reaches [[15/8]] in 11 generators, entirely contained within the 12-tone well temperament, and also [[7/5]] in 18 generators.
In the 7-limit, temperament reaches [[15/8]] in 11 generators, entirely contained within the 12-tone well temperament, and also [[7/5]] in 18 generators.
[[Category: Rank 2]]
 
== Temperament data ==
== Temperament data ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 67108864/66976875, {{monzo| -13 3 -17 17 }}


Comma list: 67108864/66976875, 6281023877654589/6250000000000000
{{Mapping|legend=1| 1 10 -7 8 | 0 -102 113 131 }}


{{mapping|legend=1| 1 10 -7 8 | 0 -102 113 131 }}
: mapping generators: ~2, 160000/151263


: mapping generators: ~2 = 1\1, 160000/151263 = 98.998
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, 160000/151263 = 98.9982


[[Support]]ing [[ET]]s: {{EDOs|97, 400}}, ...
{{Optimal ET sequence|legend=1| 97, 303, 400, 1297, 1697c }}
 
[[Badness]]: 1.10


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 166698/166375, 422576/421875, 67108864/66976875
Comma list: 166698/166375, 422576/421875, 67108864/66976875


{{mapping|legend=1| 1 10 -7 8 16 | 0 -102 113 131 -152 }}
Mapping: {{mapping| 1 10 -7 8 16 | 0 -102 113 131 -152 }}


: mapping generators: ~2 = 1\1, 69120/65219 = 98.998
Optimal tuning (CTE): ~2 = 1\1, 69120/65219 = 98.9994


[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ...
Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}
 
Badness: 0.251


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 4096/4095, 105644/105625, 166698/166375


Comma list: 676/675, 4096/4095, 166698/166375, 105644/105625
Mapping: {{mapping| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}


{{mapping|legend=1| 1 10 -7 8 16 7 | 0 -102 113 131 -152 -40 }}
Optimal tuning (CTE): ~2 = 1\1, 55/52 = 98.9992


: mapping generators: ~2 = 1\1, 55/52 = 98.998
Optimal ET sequence: {{Optimal ET sequence| 97, 303, 400 }}


[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ...
Badness: 0.126


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500
Comma list: 676/675, 4096/4095, 11016/11011, 14400/14399, 93639/93500


{{mapping|legend=1| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}
Mapping: {{mapping| 1 10 -7 8 16 7 -12 | 0 -102 113 131 -152 -40 195 }}
 
Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993


: mapping generators: ~2 = 1\1, 55/52 = 98.998
Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}


[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ...
Badness: 0.0776


=== 19-limit ===
=== 19-limit ===


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399
Comma list: 676/675, 2926/2925, 4096/4095, 6175/6174, 11016/11011, 14400/14399


{{mapping|legend=1| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3}}
Mapping: {{mapping| 1 10 -7 8 16 7 -12 4 | 0 -102 113 131 -152 -40 195 3 }}
 
Optimal tuning (CTE): ~2 = 1\1, 18/17 = 98.9993
 
Optimal ET sequence: {{Optimal ET sequence| 97g, 303g, 400 }}


: mapping generators: ~2 = 1\1, 55/52 = 98.998
Badness: 0.0541


[[Support]]ing [[ET]]s: {{EDOs|97, 303, 400}}, ...
[[Category:Gregorian leap day| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Gregorian_leap_day"